Subatomic Physics

238 Angular Momentum and Isospin
8.12. ∗ Compare ∆ECoul for A =10, 80, and 200. Why is it more difficult (or
impossible) to find all the members of an isospin multiplet in heavy nuclei
than in light nuclei?
8.13. Consider the reactions
γA −→ nA ′
dA −→ pA ′
dA −→ αA ′
3 HeA ′′ −→ 3 HA ′ .
If A is a self-conjugate (N = Z) nuclide, what are the isospin states in A ′ that
can be reached by these reactions? The photon “carries” isospin 0 and 1. If
A ′′ has isospin 0, or 1 3
2 ,or2 , what are the possible values of the isospin states
in A ′ ?
8.14. (a) Prove the commutation relations
[I±,I 2 ]=0, [I3,I±] =±I±, [I+,I−] =2I3.
(b) Use these commutation relations and Eq. (8.24) to prove Eq. (8.27).
8.15. (a) Use the generalization of Eq. (8.30) to deduce the strangeness content of
the D 0 meson of isospin 1/2, the ηc meson of isospin 0, the Λ + c baryon
of isospin 0. Assume that B = T =0.
(b) Repeat part (a) for bottomness content for the B− of isospin 1
2 ,ifC =
T =0.
8.16. The angular distribution of neutral pions produced in the reaction np → dπ 0
is found to be (almost) symmetrical about 90 ◦ in the c.m. (see A.K. Opper et
al., Phys. Rev. Lett. 91, 212302 (2003)) Show that this follows from isospin
conservation.
8.17. Projection operators have the properties that P |a〉 and P |b〉 =0if〈a|b〉 =0
and P 2 = P . In terms of the isospin operator I3,op determine a projection
operator Pp for the proton and Pn for the neutron, such that Pp|p〉 =1,
Pp|n〉 =0,Pn|n〉 =1,Pn|p〉 =0. (For isospin 1/2 I 2 3,op =1/4).